Definition 4.3 (see [1]})  A set \(A\subset \mathbb {R}^n\)  is said to be inner \(\alpha \) -thin
at infinity if any of (equivalent) assertions (i)–(v) in Theorem REF  holds true. Thus \(A\)  is inner \(\alpha \) -thin at infinity if and only if some (equivalently, every) point \(y\in \mathbb {R}^n\)  is inner \(\alpha \) -irregular for \(A_y^*\) , the inverse of \(A\)  with respect to the sphere \(S(y,1)\) .The concept of inner \(\alpha \) -thinness of a set at infinity, and particularly its alternative characterization provided by Theorem REF (iii), has already found a number of applications to minimum \(\alpha \) -Riesz energy problems for condensers, see e.g. [2]}, [3]}–[4]}.

